Question

What is the molar mass of a gas which has a density of 5.75 g/L at 25.0 C and 1.00 atm pressure?

Answer 1

Let's start with the two following equations and see what we can do here to get to a form we want:\[PV=nRT\]\[D=\frac{m}{V}\]We can also add another equation to the mix because we know that if we have a mass of a substance as well as a molar mass we can find the amount of substance:\[n=\frac{m}{MM}\]Let's rearrange this equation to solve for m, which we can shove into the density equation:\[m=(n)(MM)\]Now, if we substitute this into the density equation, we get:\[D=\frac{(n)(MM)}{V}\]Now, let's solve this for n, which we can shove into the ideal gas equation:\[n=\frac{DV}{MM}\]Now, finally let's plug this value into the ideal gas equation to get:\[PV=nRT \rightarrow PV=\frac{DVRT}{MM}\]The volumes on either side of the equation will cancel to give:\[P=\frac{DRT}{MM}\]We know pressure, density, the gas constant, and temperature, so we can solve for molar mass of the substance:\[MM=\frac{DRT}{P} \rightarrow MM=\frac{(5.75\frac{g}{L})(0.08206\frac{L \times atm}{K \times mol})(298K)}{(1.00atm)}=141\frac{g}{mol}\]And you'll notice that the units match up for a molar mass.